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- Question about a mathematical detail in the proof of the thermodynamic stability relation
I am watching Kardar's Statistical Mechanics course in my spare time and I am struggling to understand a mathematical detail in the proof of the thermodynamic stability condition. See Eq. I.62 here.
The author considers a homogeneous system at equilibrium with intensive and extensive variables (T,J,\mu) and (E,x,N), respectively. Then, He imagines that the latter system is divided into two subsystems (A and B) that can exhange energy. Under the assumption that E,x and N are conserved, we have \delta E_A=-\delta E_B, \delta x_A=-\delta x_B and \delta N_A=-\delta N_B. What is not clear to me is the statement "Since the intensive variables are themselves functions of the extensive coordinates, to first order in the variations of (E, x, N), we have \delta T_A=-\delta T_B, \delta J_A=-\delta J_B and \delta \mu_A=-\delta \mu_B."
Can anyone explain to me the previous statement more in detail?
Thank you!
The author considers a homogeneous system at equilibrium with intensive and extensive variables (T,J,\mu) and (E,x,N), respectively. Then, He imagines that the latter system is divided into two subsystems (A and B) that can exhange energy. Under the assumption that E,x and N are conserved, we have \delta E_A=-\delta E_B, \delta x_A=-\delta x_B and \delta N_A=-\delta N_B. What is not clear to me is the statement "Since the intensive variables are themselves functions of the extensive coordinates, to first order in the variations of (E, x, N), we have \delta T_A=-\delta T_B, \delta J_A=-\delta J_B and \delta \mu_A=-\delta \mu_B."
Can anyone explain to me the previous statement more in detail?
Thank you!